Incomparable families and maximal trees
Volume 234 / 2016
Fundamenta Mathematicae 234 (2016), 73-89
MSC: 03E17, 03E35, 03G05.
DOI: 10.4064/fm125-1-2016
Published online: 20 January 2016
Abstract
We answer several questions of D. Monk by showing that every maximal family of pairwise incomparable elements of $\mathcal P(\omega )/\mathit {fin}$ has size continuum, while it is consistent with the negation of the Continuum Hypothesis that there are maximal subtrees of both $\mathcal P(\omega )$ and $\mathcal P(\omega )/\mathit {fin}$ of size $\omega _1$.
